3.155 \(\int \frac {C(b x)^2}{x^8} \, dx\)

Optimal. Leaf size=259 \[ \frac {1}{168} \pi ^3 b^7 \text {Int}\left (\frac {C(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{x},x\right )+\frac {2}{315} \sqrt {2} \pi ^3 b^7 S\left (\sqrt {2} b x\right )+\frac {\pi ^3 b^7 S\left (\sqrt {2} b x\right )}{72 \sqrt {2}}+\frac {\pi ^2 b^6}{336 x}-\frac {b C(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{21 x^6}-\frac {b^2}{210 x^5}-\frac {b^2 \cos \left (\pi b^2 x^2\right )}{210 x^5}+\frac {67 \pi ^2 b^6 \cos \left (\pi b^2 x^2\right )}{5040 x}+\frac {\pi ^2 b^5 C(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{168 x^2}+\frac {13 \pi b^4 \sin \left (\pi b^2 x^2\right )}{2520 x^3}+\frac {\pi b^3 C(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{84 x^4}-\frac {C(b x)^2}{7 x^7} \]

[Out]

-1/210*b^2/x^5+1/336*b^6*Pi^2/x-1/210*b^2*cos(b^2*Pi*x^2)/x^5+67/5040*b^6*Pi^2*cos(b^2*Pi*x^2)/x-1/21*b*cos(1/
2*b^2*Pi*x^2)*FresnelC(b*x)/x^6+1/168*b^5*Pi^2*cos(1/2*b^2*Pi*x^2)*FresnelC(b*x)/x^2-1/7*FresnelC(b*x)^2/x^7+1
/84*b^3*Pi*FresnelC(b*x)*sin(1/2*b^2*Pi*x^2)/x^4+13/2520*b^4*Pi*sin(b^2*Pi*x^2)/x^3+67/5040*b^7*Pi^3*FresnelS(
b*x*2^(1/2))*2^(1/2)+1/168*b^7*Pi^3*Unintegrable(FresnelC(b*x)*sin(1/2*b^2*Pi*x^2)/x,x)

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Rubi [A]  time = 0.24, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\text {FresnelC}(b x)^2}{x^8} \, dx \]

Verification is Not applicable to the result.

[In]

Int[FresnelC[b*x]^2/x^8,x]

[Out]

-b^2/(210*x^5) + (b^6*Pi^2)/(336*x) - (b^2*Cos[b^2*Pi*x^2])/(210*x^5) + (67*b^6*Pi^2*Cos[b^2*Pi*x^2])/(5040*x)
- (b*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(21*x^6) + (b^5*Pi^2*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(168*x^2) - F
resnelC[b*x]^2/(7*x^7) + (b^7*Pi^3*FresnelS[Sqrt[2]*b*x])/(72*Sqrt[2]) + (2*Sqrt[2]*b^7*Pi^3*FresnelS[Sqrt[2]*
b*x])/315 + (b^3*Pi*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(84*x^4) + (13*b^4*Pi*Sin[b^2*Pi*x^2])/(2520*x^3) + (b^
7*Pi^3*Defer[Int][(FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/x, x])/168

Rubi steps

\begin {align*} \int \frac {C(b x)^2}{x^8} \, dx &=-\frac {C(b x)^2}{7 x^7}+\frac {1}{7} (2 b) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{x^7} \, dx\\ &=-\frac {b^2}{210 x^5}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{21 x^6}-\frac {C(b x)^2}{7 x^7}+\frac {1}{42} b^2 \int \frac {\cos \left (b^2 \pi x^2\right )}{x^6} \, dx-\frac {1}{21} \left (b^3 \pi \right ) \int \frac {C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^5} \, dx\\ &=-\frac {b^2}{210 x^5}-\frac {b^2 \cos \left (b^2 \pi x^2\right )}{210 x^5}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{21 x^6}-\frac {C(b x)^2}{7 x^7}+\frac {b^3 \pi C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{84 x^4}-\frac {1}{168} \left (b^4 \pi \right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^4} \, dx-\frac {1}{105} \left (b^4 \pi \right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^4} \, dx-\frac {1}{84} \left (b^5 \pi ^2\right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{x^3} \, dx\\ &=-\frac {b^2}{210 x^5}+\frac {b^6 \pi ^2}{336 x}-\frac {b^2 \cos \left (b^2 \pi x^2\right )}{210 x^5}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{21 x^6}+\frac {b^5 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{168 x^2}-\frac {C(b x)^2}{7 x^7}+\frac {b^3 \pi C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{84 x^4}+\frac {13 b^4 \pi \sin \left (b^2 \pi x^2\right )}{2520 x^3}-\frac {1}{336} \left (b^6 \pi ^2\right ) \int \frac {\cos \left (b^2 \pi x^2\right )}{x^2} \, dx-\frac {1}{252} \left (b^6 \pi ^2\right ) \int \frac {\cos \left (b^2 \pi x^2\right )}{x^2} \, dx-\frac {1}{315} \left (2 b^6 \pi ^2\right ) \int \frac {\cos \left (b^2 \pi x^2\right )}{x^2} \, dx+\frac {1}{168} \left (b^7 \pi ^3\right ) \int \frac {C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x} \, dx\\ &=-\frac {b^2}{210 x^5}+\frac {b^6 \pi ^2}{336 x}-\frac {b^2 \cos \left (b^2 \pi x^2\right )}{210 x^5}+\frac {67 b^6 \pi ^2 \cos \left (b^2 \pi x^2\right )}{5040 x}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{21 x^6}+\frac {b^5 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{168 x^2}-\frac {C(b x)^2}{7 x^7}+\frac {b^3 \pi C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{84 x^4}+\frac {13 b^4 \pi \sin \left (b^2 \pi x^2\right )}{2520 x^3}+\frac {1}{168} \left (b^7 \pi ^3\right ) \int \frac {C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x} \, dx+\frac {1}{168} \left (b^8 \pi ^3\right ) \int \sin \left (b^2 \pi x^2\right ) \, dx+\frac {1}{126} \left (b^8 \pi ^3\right ) \int \sin \left (b^2 \pi x^2\right ) \, dx+\frac {1}{315} \left (4 b^8 \pi ^3\right ) \int \sin \left (b^2 \pi x^2\right ) \, dx\\ &=-\frac {b^2}{210 x^5}+\frac {b^6 \pi ^2}{336 x}-\frac {b^2 \cos \left (b^2 \pi x^2\right )}{210 x^5}+\frac {67 b^6 \pi ^2 \cos \left (b^2 \pi x^2\right )}{5040 x}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{21 x^6}+\frac {b^5 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{168 x^2}-\frac {C(b x)^2}{7 x^7}+\frac {b^7 \pi ^3 S\left (\sqrt {2} b x\right )}{72 \sqrt {2}}+\frac {2}{315} \sqrt {2} b^7 \pi ^3 S\left (\sqrt {2} b x\right )+\frac {b^3 \pi C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{84 x^4}+\frac {13 b^4 \pi \sin \left (b^2 \pi x^2\right )}{2520 x^3}+\frac {1}{168} \left (b^7 \pi ^3\right ) \int \frac {C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x} \, dx\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 0, normalized size = 0.00 \[ \int \frac {C(b x)^2}{x^8} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[FresnelC[b*x]^2/x^8,x]

[Out]

Integrate[FresnelC[b*x]^2/x^8, x]

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fricas [A]  time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\rm fresnelc}\left (b x\right )^{2}}{x^{8}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnelc(b*x)^2/x^8,x, algorithm="fricas")

[Out]

integral(fresnelc(b*x)^2/x^8, x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\rm fresnelc}\left (b x\right )^{2}}{x^{8}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnelc(b*x)^2/x^8,x, algorithm="giac")

[Out]

integrate(fresnelc(b*x)^2/x^8, x)

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maple [A]  time = 0.01, size = 0, normalized size = 0.00 \[ \int \frac {\FresnelC \left (b x \right )^{2}}{x^{8}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(FresnelC(b*x)^2/x^8,x)

[Out]

int(FresnelC(b*x)^2/x^8,x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\rm fresnelc}\left (b x\right )^{2}}{x^{8}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnelc(b*x)^2/x^8,x, algorithm="maxima")

[Out]

integrate(fresnelc(b*x)^2/x^8, x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {FresnelC}\left (b\,x\right )}^2}{x^8} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(FresnelC(b*x)^2/x^8,x)

[Out]

int(FresnelC(b*x)^2/x^8, x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {C^{2}\left (b x\right )}{x^{8}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnelc(b*x)**2/x**8,x)

[Out]

Integral(fresnelc(b*x)**2/x**8, x)

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